其他名称：
24-21-00245
项目负责人：
Ratanov Nikita
发表日期：
2024
主持机构：
Federal State Budgetary Educational Institution of Higher Education "Chelyabinsk State University",
国家：
俄罗斯
开始日期：
2024
结束日期：
2025
简介：
Classical stochastic analysis based on Ito integrals w.r.t. the Wiener process was created in the middle of the 20th century in the works of S.N. Bernshtein, I.I. Gikhman, R.L. Stratonovich and K. Ito, which gave rise to the theory of ordinary stochastic differential equations and stochastic partial differential equations. The subsequent development of stochastic analysis, its generalisation and improvement is carried out on the basis of Levy processes, as well as processes with infinitesimal generator of a fractional order, and involves more and more sophisticated mathematical constructions. Meanwhile, some interesting aspects of stochastic analysis based on processes with alternating states, as well as those associated with processes of finite variation, have been largely left out. It is very important to develop the theory of stochastic differential equations both on the basis of Markov- and non-Markov-modulated processes.$This theory was previously substantially developed in the works of numerous scientists. The results of these efforts were recently summed up in the monograph N.Ratanov, A.D.Kolesnik. Telegraph Processes and Option Pricing. 2ed, Springer -- Heidelberg, New York, Dordrecht, London, 2022. However, despite all efforts, this direction has not yet received due development. In particular, the study of such processes with random alternating velocities, with jumps of random amplitude and/or random intensity, has just begun and is far from complete.$The project is aimed at creating and developing new mathematical methods and ideas for the “new stochastic analysis”, the need for which has already matured. New classes of stochastic differential equations based on piecewise deterministic processes will be studied. One of the main issues to be considered within the framework of the project is the study of explicit representations of solutions to a wide class of differential equations based on such processes, the corresponding Cauchy problems, as well as boundary value problems. Distributions of functionals from solutions will be studied, such as the time of the first crossing a given threshold (and a two-threshold problem), the number of crossings of this level, local time, exponential functional, and others.$At the second stage of the project, a stochastic analysis will be built on the basis of processes with alternating random parameters, which is a natural continuation of the results planned for the first stage. First of all, (inhomogeneous) compound Poisson processes with random intensities and their integrals will be considered from this point of view.$The problem of the existence of martingales will be studied.$ Numerous applications of both classical stochastic analysis and its various generalizations have significant drawbacks associated, for example, with the unlimited variation of the Wiener process, the infinite propagation velocity of perturbations, and the total non-differentiability of its trajectories a.s. In view of this circumstance, one should pay attention to the very promising prospects and possibilities for applications of processes with alternating states and finite variation processes in various areas traditional for the application of diffusion processes. The project involves not only the study of new classes of stochastic differential equations, but also a whole range of applied problems based on these stochastic equations. First of all, a series of studies aimed at building new models of financial markets and financial instruments, including exotic options, is envisaged. Particular attention will be paid to barrier options based on piecewise deterministic processes and processes with alternating random parameters. In addition, other types of applied problems will be studied, such as models for describing neuronal activity.$The project continues and develops the main research trends of recent years. The project continues and develops the main areas of research in recent years. The experience possessed by the participants allows us to solve the tasks.
专业领域：
其他
语种：
英语
